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MR Image Denoising and Super-Resolution Using Regularized Reverse Diffusion

Hyungjin Chung, Eun Sun Lee, Jong Chul Ye

arXiv:2203.12621v1eess.IVcs.AIcs.CVcs.LG

TL;DR

MRI denoising must handle complex noise and distribution shifts while avoiding the blur associated with MMSE estimates. The paper uses score-based reverse diffusion, beginning from noisy images, and reuses the score function for super-resolution. Experiments report strong denoising and resolution results together with controllable sampling and uncertainty quantification.

  • Problem

    MRI denoisers can produce blurred outputs and degrade under distribution shifts or noise distributions that differ from their assumptions.

  • Method

    The method hijacks score-based reverse diffusion by starting from noisy-image distributions and uses the same score function to super-resolve denoised images.

  • Results

    The method achieves state-of-the-art SNR and CNR, outperforming comparison methods by a large margin.

  • Takeaways & Limitations

    Reverse-diffusion sampling provides flexible denoising control and uncertainty quantification while preserving sharp image detail.

Abstract

from arXiv · show

Patient scans from MRI often suffer from noise, which hampers the diagnostic capability of such images. As a method to mitigate such artifact, denoising is largely studied both within the medical imaging community and beyond the community as a general subject. However, recent deep neural network-based approaches mostly rely on the minimum mean squared error (MMSE) estimates, which tend to produce a blurred output. Moreover, such models suffer when deployed in real-world sitautions: out-of-distribution data, and complex noise distributions that deviate from the usual parametric noise models. In this work, we propose a new denoising method based on score-based reverse diffusion sampling, which overcomes all the aforementioned drawbacks. Our network, trained only with coronal knee scans, excels even on out-of-distribution in vivo liver MRI data, contaminated with complex mixture of noise. Even more, we propose a method to enhance the resolution of the denoised image with the same network. With extensive experiments, we show that our method establishes state-of-the-art performance, while having desirable properties which prior MMSE denoisers did not have: flexibly choosing the extent of denoising, and quantifying uncertainty.

I. INTRODUCTION

MRI denoising is challenging because real scans can contain complex noise, distribution shifts, and blur from MMSE-based estimates. The paper proposes score-based reverse diffusion to denoise and super-resolve images while enabling controllable sampling and uncertainty quantification.

  • Complex MRI noise can obscure clinically relevant image details, making denoising important for diagnosis.
  • Deep denoisers can degrade under test-distribution shifts or nonparametric noise, while MMSE estimates often produce blurrier outputs.
  • The proposed method starts reverse diffusion from noisy-image distributions rather than pure Gaussian noise.
  • The same score function is used to super-resolve denoised images, retaining high-frequency information and producing sharper results.
  • Reverse diffusion is used to provide fine-grained denoising control and posterior samples for uncertainty quantification.

A. Score-based diffusion model

The score-based diffusion model transforms data toward Gaussian noise through a forward SDE and uses a learned score function to reverse that process. The score is trained with denoising score matching and can be used with numerical reverse-SDE samplers.

  • The forward diffusion process transforms the data distribution toward an approximately isotropic Gaussian distribution over time.
  • The model uses a variance-exploding SDE with linear drift and scalar diffusion coefficients.
  • The reverse SDE incorporates the score function, the gradient of the log probability, to run diffusion backward in time.
  • A time-conditional neural network estimates the score using denoising score matching, which supports scalable training.
  • The trained score can be used with Euler–Maruyama, higher-order solvers, or predictor-corrector samplers.

B. Come-Closer-Diffuse-Faster (CCDF) [24]

CCDF reduces diffusion sampling cost by starting from a corrupted measurement and applying only the final reverse-diffusion steps. Its stochastic-contraction view explains why this shortened process can converge under suitable data-consistency mappings.

  • Standard diffusion sampling is slow because it starts from Gaussian noise and may require thousands of neural-network evaluations.
  • CCDF instead forward-diffuses the corrupted image and uses only the last few reverse-diffusion steps.
  • The reverse diffusion process is treated as a stochastic contraction mapping when paired with a non-expansive data-consistency mapping.
  • The noisy-measurement model represents the observation as a clean image plus a Gaussian noise vector.
  • Tweedie’s formula yields the posterior mean, corresponding to the MMSE estimator, when the correct score function is available.

III. MAIN CONTRIBUTIONS

The paper connects score-based denoising with reverse diffusion and addresses the blur of single-step MMSE estimates. It also estimates noise parameters nonparametrically and supports controllable denoising through reverse-diffusion sampling.

  • The diffusion time index is discretized into N uniform intervals for the noise schedule and sampled process.
  • The forward-diffusion formulation matches the noisy-image model when the observation noise scale equals the diffusion noise scale.
  • The trained score can denoise a corrupted image in one step through Tweedie’s formula when its noise variance lies within the training schedule.
  • Single-step MMSE denoising often produces blurred results, especially as the noise scale increases.
  • The proposed reverse-diffusion approach is motivated by the need for fine-grained control and posterior sampling to obtain sharper, higher-fidelity estimates.

B. Reverse diffusion as denoiser

The method treats a noisy image as an intermediate diffusion sample and runs reverse diffusion from its estimated noise level. The denoising extent is adjustable, enabling either stronger noise reduction or greater detail preservation.

  • Reverse diffusion as denoiser: The reverse SDE solver can generate a sample path from an initial noisy image rather than pure Gaussian noise.The noisy image is treated as a sample from an intermediate distribution, and reverse diffusion begins at the corresponding time.
  • Reverse diffusion as denoiser: Noise-level estimation selects the reverse-diffusion starting point when the true noise level is unknown.The proposed estimator uses eigenvalue analysis of the covariance matrix, although other estimators are possible.
  • Reverse diffusion as denoiser: The denoiser performs a reduced number of reverse-diffusion iterations determined by the estimated time and control parameter α.Algorithm 2 sets t′ from the estimated noise level and uses N′ = αt′N iterations.
  • Reverse diffusion as denoiser: Clinicians can vary α to trade off aggressive denoising against preservation of original image details.Multiple α values can be tested to select the best-performing denoising strength.

C. Low frequency regularizer

The low-frequency regularizer stabilizes stochastic reverse diffusion by preserving low-frequency structure from the initial noisy image. It alternates diffusion with data consistency and can reach a fixed point through stochastic contraction.

  • Low frequency regularizer: The regularizer addresses structural alterations that may arise from stochastic reverse diffusion in medical images.The method is motivated by preventing altered structure during denoising.
  • Low frequency regularizer: The method preserves low-frequency components while primarily correcting high-frequency components associated with noise.This design keeps important structure intact while focusing denoising on higher-frequency content.
  • Low frequency regularizer: The regularization operator combines the current estimate with the initial image’s low-frequency component using λ ∈ [0, 1].The Fourier-domain formulation uses F and PΩ to select the low-frequency region and λ to control regularization strength.
  • Low frequency regularizer: Reverse diffusion and data consistency can be applied alternately to reach a fixed point through stochastic contraction.The data-consistency operator is described as non-expansive for the normalized filter.
  • Low frequency regularizer: R2D2+1 combines the super-resolution process with forward-diffusion initialization and uses few tens of iterations rather than a few thousand.The process starts from x̂M instead of random Gaussian noise.

IV. METHODS

The experiments train a score function on fully sampled fastMRI knee magnitude images and test it on 28 high-flip-angle, reduced-FOV liver MRI volumes. The implementation uses VE-SDE training with an NCSN++ network and predictor-corrector sampling.

  • Experimental Data: The score network is trained on 320×320 fully sampled single-coil MRI magnitude images from the fastMRI knee dataset.The training images themselves contain noise from the simulated single-coil measurement.
  • Experimental Data: The liver test set contains 28 three-dimensional volumes acquired with a Siemens Skyra scanner using high-flip-angle CAIPIRINHA and 1.5 mm slices.The reduced field of view and thin slices were used to capture small lesions.
  • Implementation: Training uses the VE-SDE objective with σmin = 0.01, σmax = 378, batch size 2, and Adam optimization.The learning rate warms up linearly for 5000 steps to 2e-4 and then remains static.
  • Implementation: The score model uses the NCSN++ architecture with Fourier-feature time conditioning and a predictor-corrector sampler.The low-frequency regularization parameter is λ = 0.005 and α is fixed at 0.2 unless otherwise specified.

C. Comparison methods

The study compares the proposed method with BM3D and self-supervised denoisers using SNR and CNR, with statistical significance assessed by repeated-measures ANOVA and Bonferroni correction. Standard PSNR and SSIM cannot be computed because clean ground-truth liver data are unavailable.

  • Comparison methods: BM3D is the representative non-deep-learning baseline and uses estimated noise variance with Wiener filtering and hard thresholding.The implementation uses official Python defaults.
  • Comparison methods: The deep-learning comparisons include Noise2Noise, Neighbor2Neighbor, and Noise2Score because clean liver data are unavailable.Supervised and cycleGAN-based approaches are excluded for lacking appropriate clean liver data.
  • Quantitative evaluation and statistical analysis: SNR and CNR quantify improvement in signal quality and contrast using signal-dominant and background regions of interest.The ROI definitions use means and signal standard deviations from circular masked regions.
  • Quantitative evaluation and statistical analysis: SNR evaluation uses eight liver-segment regions per volume, while CNR evaluation uses two major liver vessels.Circular disks with 10 mm radius are used for the SNR regions.
  • Quantitative evaluation and statistical analysis: PSNR and SSIM are unavailable because the experiment lacks clean ground-truth liver images.The evaluation therefore relies on SNR and CNR instead of standard full-reference metrics.
  • Quantitative evaluation and statistical analysis: Repeated-measures ANOVA with post-hoc Bonferroni correction assesses statistical relevance across methods.The pairwise comparisons reported in Table II include Input and BM3D.

V. RESULTS

Across liver conditions, the proposed method removes noise while sharpening images, achieves the strongest reported SNR and CNR, and supports adjustable denoising and uncertainty estimation. Regularization preserves important structures and suppresses artifacts, while denoising strength must match the noise pattern.

  • In vivo study: The proposed method was the only approach reported to remove noise while sharpening images across all liver conditions.The comparison covered early cirrhosis with HCC, normal liver, and advanced cirrhosis.
  • In vivo study: The proposed method achieved the highest scores for both SNR and CNR, whereas competing outputs tended to be washed out with unclear boundaries and vessel structures.N2N and N2Score outperformed BM3D but remained substantially behind R2D2+, particularly in CNR.
  • Flexibility and uncertainty: Higher α improved denoising for broadly distributed, near-Gaussian noise, while lower α better preserved liver texture and details under strongly non-Gaussian noise.The latter case included spatially concentrated pulsation artifact and high-scale central noise.
  • Flexibility and uncertainty: Five posterior samples enabled mean and standard-deviation maps, with localized high variance indicating regions where clinicians should make decisions conservatively.The first sample showed no excessive regional variance, whereas the second contained higher-variance regions.
  • Impact of low-frequency regularization: Low-frequency regularization conserved vessel and other important structures while eliminating shaggy-looking structured artifacts caused by unregularized sampling.The scheme was introduced because stochastic reverse diffusion could otherwise alter medical-image structure.

B. Ablation study on super-resolution

The ablation compares reconstruction quality with and without post-hoc resolution enhancement and examines related regularization and uncertainty properties. The enhancement improves resolution and CNR while slightly decreasing SNR, while the broader method supports uncertainty estimation and flexible denoising.

  • B. Ablation study on super-resolution: The post-hoc super-resolution step greatly enhances resolution and CNR, at the expense of a slight decrease in SNR.The authors conclude that the proposed method has an overall positive effect on reconstruction.
  • B. Ablation study on super-resolution: The method is reported to achieve state-of-the-art SNR and CNR while outperforming comparison methods by a large margin.The conclusion attributes this performance to reverse diffusion and non-expansive mapping regularization, together with the proposed resolution enhancement.
  • B. Ablation study on super-resolution: Low-frequency regularization preserves vessel structure and eliminates structured shaggy artifacts observed without regularization.The ablation compares reconstructions with and without regularization against noisy images.
  • B. Ablation study on super-resolution: The method estimates uncertainty by sampling multiple posterior reconstructions and computing statistics from those samples.This provides uncertainty quantification for the inverse problem rather than relying on a single reconstruction.

APPENDIX

Additional experiments compare the proposed method with BM3D, Noise2Noise, and Noise2Score. The reported comparison emphasizes sharper vessel structures and better preservation of high-frequency detail.

  • ADDITIONAL EXPERIMENTAL RESULTS: The proposed method is compared extensively with BM3D, Noise2Noise, and Noise2Score in additional experiments.The comparison is presented in Fig. 7.
  • ADDITIONAL EXPERIMENTAL RESULTS: The proposed method reconstructs sharp vessel structures while removing excessive noise around them, whereas the comparison methods preserve less high-frequency detail.The passage states that the proposed method clearly outperforms all compared methods in the shown results.
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